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degrees, algebraists struggled for nearly three centuries to pro-
duce a similar algebraic solution for the general quintic
ax* + to* + ax* + ^2 + ea!+/= 0.

They failed. It is here that Abel enters.
The following extracts are given partly to show how a great
inventive mathematician thought and partly for their intrinsic
interest. They are from Abel's memoir On the algebraic resolution
of equations.
'One of the most interesting problems of algebra is that of the
algebraic solution of equations. Thus we find that nearly all
mathematicians of distinguished rank have treated this subject.
We arrive without difficulty at the general expression of the
roots of equations of the first four degrees. A uniform method
for solving these equations was discovered and it was believed
to be applicable to an equation of any degree; but in spite of all
the efforts of Lagrange and other distinguished mathematicians
the proposed end was not reached. That led to the presumption
that the solution of general equations was impossible algebrai-
cally; but this is what could not be decided, since the method
followed could lead to decisive .conclusions only in the case
where the equations were solvable. In effect they proposed to
solve equations without knowing whether it was possible. In
this way one might.indeed arrive at a solution, although that
was by no means certain; but if by ill luck the solution was
impossible, one might seek it for an eternity, without finding it.
To arrive infallibly at something hi this matter, we must there-
. fore follow another road. We can give the problem such a form
that it shall always be possible to solve it, as we can always do
with any problem.* Instead of asking for a relation of which it
is not known whether it exists or not, we must ask whether
such a relation is indeed possible. , . . When a problem is posed
in this way, the very statement contains the germ of the solu-
tion and indicates what road must be taken; and I believe there
* '. . . ce gut on peut toujours faire tfun probleme quelconqw? is what
Abel says. This seems a trifle too optimistic; at least for ordinary
mortals. How would the method be applied to Fermat's Last
* 341